Complexity of Homogeneous Co-Boolean Constraint Satisfaction Problems

Constraint Satisfaction Problems (CSP) constitute a convenient and uniform framework to describe many algorithmic and combinatorial problems from graph theory, artificial intelligence, optimization, computational molecular biology, etc. The general CSP problem is well-known to be NP-complete. However, we can consider the parametric version of the CSP problem, denoted CSP(S), where the template S is a set of allowed relations upon which any instance of the problem is constructed. The goal is to study the complexity of the parametric CSP, recognizing the conditions allowing us to distinguish between tractable and intractable instances of the considered problem, as well as the understanding of the complexity classes to which these instances belong. The study of computational complexity of constraint satisfaction problems was started by Schaefer in his landmark paper [7], where he completely characterized the complexity of Boolean CSP, distinguishing between polynomial and NP-complete instances. Feder and Vardi [4] extended this study to constraint satisfaction problems over finite domains, for which they conjectured the existence of a Dichotomy Theorem. So far, this claim was proved only for the ternary domain by Bulatov [2], exhibiting an involved Dichotomy Theorem, whereas the claim remains open for higher cardinality domains.

A fundamental result from Feder, Madelaine and Stewart [3] shows that for every set of relations S, there exists a set F of unary functions, such that the problems CSP(S) and CSP(F • ) are polynomial-time equivalent, where F • is the set of the graphs of functions from F . Thus, CSPs over unary functions are as powerful as general CSP problems. Graphs of unary functions give us a very structural template which is really convenient to work with.

In this paper, we focus on templates built upon homogeneous co-Boolean functions on a domain D, that is, unary functions sharing a range of size two. By convention, we take the range {0, 1} ⊆ D. The goal of this paper is more to present well-known results from another angle and initiate a new way to study the complexity of CSP(S) problems rather than to present new polynomialtime algorithms for CSP. The paper is organized as follows. The first section describes general notions used in this paper. Then we introduce the parametric CSP problem in general and more specifically on graphs of homogeneous co-Boolean functions, as well as some intermediary results. In the last section, we show a dichotomy theorem for every finite domain D of CSP built upon graphs of homogeneous co-Boolean functions. The paper terminates with some concluding remarks.

Let f : D → D be a unary function over a finite domain D = {0, . . . , n -1}. This function f is called co-Boolean if the range of f , also named the co-domain, is of size 2. In this paper, we focus on homogeneous co-Boolean functions, i.e. co-Boolean functions sharing the same co-domain. By convention, we choose {0, 1} ⊆ D to be this shared co-domain. Since in this short note we deal with homogeneous co-Boolean functions only, we can simply named these functions “co-Boolean functions” without any confusions. The idea behind co-Boolean functions is a partition of the domain D into two disjoint sub-domains, where f acts as a characteristic function.

Since we study in this paper only unary functions, each function will be considered to be unary even if we do not explicitly mention its arity. We assume that the domain D is ordered by an arbitrary but fixed total order <. Without loss of generality, we can assume that < is the natural order 0

In other words, the algebraic structure (D; <) is a chain.

The graph of a function f is the binary relation

Given a tuple t in a ℓ-ary relation R, we denote by t[i] the i-th coordinate of t, with 1 ≤ i ≤ ℓ. We say that a relation R is closed under (or preserved by) a k-ary operation p, or that p is a polymorphism of R, if for any choice of not necessarily distinct k tuples t 1 , . . . , t k ∈ R the following membership condition holds:

i.e., that the new tuple constructed coordinate-wise from t 1 , . . . , t k by means of p belongs to R. We denote by Pol R the set of polymorphisms of a relation R and by Pol S the polymorphisms of every relation R in S. Recall that Pol S = R∈S Pol R.

In particular, we need to study the closure under four operations, namely majority, minority, maximum, and minimum. Maximum and minimum are binary operations, satisfying respectively the following conditions for all elements a, b ∈ D:

Both aforementioned operations are known in universal algebra as semi-lattice operations, since they correspond to the operations of join and meet. On the Boolean domain {0, 1}, the maximum operation max(x, y) translates to disjunction x ∨ y and the minumum operation min(x, y) translates to conjunction x ∧ y. More generally, a semi-lattice operation is a binary associative, commutative and idempotent operation. We say that a k-ary operation q : D k → D is

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